Question

Find the points of discontinuity of f, where f(x) = `{(sinx/x", if"  x<0),(x + 1", if"  x >= 0):}`.

Answer 1

f(x) = `{(sinx/x", if"  x<0),(x + 1", if"  x >= 0):}`

At x = 0, f(0) = 1

L.H.L. = `lim_(x->0^-)` f(x) = `lim_(h->0)(sin(-h))/-h` = 1

R.H.L. = `lim_(x->0^+)` f(x) = `lim_(h->0)` (h + 1) = 0 + 1 = 1

`lim_(x->0^-)` f(x) = `lim_(x->0^+)` f(x) = f(0)

∴ f is continuous at x = 0.

When x < 0, sin x and x are both continuous.

∴ `sinx/x` is also continuous.

When x > 0, f(x) = x + 1 is a polynomial.

∴ f is continuous.

⇒ f is not discontinuous at any point.